08+09 - Heat and Mass Transfer - in 2D

Question 1

Heat conduction

Answer 1

• Fourier’s law: q = -k.A.dT/dx
• heat flux = q’’ = q/A
• 1D steady state heat conduction: d/dx(k.A.dT/dx) +Q.A = 0
• conduction is b/ 2 different temperatures

Question 2

Essential / Natural BCs

Answer 2

• essential: where the value is specified

- natural: that will be verified automatically once the solution is found

Question 3

Convection

Answer 3

• q_in = h.A(T_inf - T)
• heat flow through fins
• b/ volume and surface of different T°

Question 4

Mass transfer

Answer 4

• J = -D.dC/dx
• dC/dt = D(∂2C/∂x2 + ∂2C/∂y2 +∂2C/∂z2)
• with advection: Gamma = -D.nabla(C) + w.c.U

Question 5

Possible BCs in heat transfer

Answer 5

• constant T°
• cstt surface heat flux
• adiabatic surface
• convection surface cdt°

Question 6

FDM for steady-state problems

Answer 6

• procedure: discretize -> energy balance method -> solving for unkn. nodal T° -> use T° field and Fourier’s law to determine heat transfer in the medium
• solution methods
• – matrix inversion: T = A-1.C (A coeff matrix, C vector of cstts)
• – Gauss-Siedel iteration: reorg to have diagonal e/ > other row e/, iterate until |T(k,i) - T(k-1,i)| < epsilon (for implicit method)

Question 7

FDM for transient problems

Answer 7

• procedure: same except solve until steady-state is reached
• eq°: 1/alpha . ∂T/∂t = nabla2(T) with alpha = k/rho.c_p thermal diffusivity
• methods
  — explicit: evaluate at p+1 with T° values from previous time step p
  time derivative is a forward difference
  stability criterion: delta(t) <= delta(x)2 /4.alpha <=> Fo <= 1/4 = FD form of Fourier number
  quicker but conditionally stable
  — implicit: evaluate each term at p+1
  time derivative is a backward difference
  nodal eq° must be solved simultaneously
  unconditionally stable but longer